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Full text of "Treatise On Analysis Vol-Ii"

12   THE SPACE L°°       175

ibr r^.kn and s^kn. For each pair (r, ,?) such that r^kn and s ^ kn ,
et A^H be the negligible set of points x e X such that \fr(x) —fs(x)\ > l/n9
ind let A be the union of the sets Ars/J (n ^ 1, r ^ &n, s ^ fcj, so that A is
i negligible set. It is clear that in X - A the sequence (/„(*)) converges
miformly to a limit /(x). The function/, which is defined almost everywhere,
s measurable (13.9.10) and bounded in X - A. Hence /e Lg (resp./e L<?).
Clearly N^C/-/,) g l/« for all r ^ fcn , and therefore / is the limit of the
;equence (/„).

13.12.5) #*/ejS?£(X, p) (where p=l or 2) and g €•&$(£,& then
^ e &%& AI), and

For fy is measurable (13.9.8.1), and |/(x)0(x)| £ |/(*)|Noo(#) almost
everywhere ; hence the result.

Remark

13.12.6) Evidently we have ^(X) c jS?J (X, ^); but in general the canon-
cal image of ^(X) in LR(X, /^) is not dense in the latter space, and L^(X, JJL)
s not separable in general (Problem 1).

'ROBLEMS

I. If A is Lebesgue measure, show that the space Lg(R, A) is not separable. (If (An) is an
infinite sequence of nonnegligible measurable sets, no two of which intersect, consider
the functions which are constant on each An and take only the values ± 1.)

I. (a) For every /x-integrable subset A of X arid every 6 > 0, let V(A, 3) denote the set of
/x-measurable real-valued functions / such that the set M of points x 6 A for which
|/(»| > 8 has measure p,(M)^8. Let «^(X, /x) denote the vector space of (finite)
real-valued /x-measurable functions on X. Show that the sets V(A, 8) form a funda-
mental system of neighborhoods of 0 for a topology on ^"(X, ft) compatible with its
vector space structure (Section 12.14, Problem 1). This topology is called the topology
of convergence in measure (with respect to ^). A sequence (/„) which tends to a limit/
in this topology is said to converge in measure to /.

(b)    Show that the intersection of all the neighborhoods V(A, 8) is the subspace ^
of negligible functions, and that the quotient space S(X, p) — «^(X, /x)/^ is metrizable.

(c)    If (/„) is any sequence in 5^(X, /x) (12.9) such that (/,) is a Cauchy sequence in
S(X, fi), show that there exists a subsequence (fnk) of (/„) such that the sequence
(fnk(x)) converges for almost all x e X. Deduce that the metrizable vector space S(X, p,)
is complete.

(d)    Every sequence (/„) of measurable real-valued functions which converges almost
everywhere to a function/ converges in measure to/.

(e)    For every finite p ^ 1, show that the space -S?&(X, p,) is dense in y(X, /x), and that
the topology induced on -S?&(X,ju.) by the topology of convergence in measure is
coarser than that defined by the semi norm Np.quality "). (Show that the set of points fo, t2) e R2